Theory of belt or rope drives
In these drives, the power transmitted depends upon the friction between the rope or belt and the rim of the pulley (denoted as sheave in American parlance).
Referring to Figure 11.1 (a), let q be the angle of wrap i. e. the angle at the pulley centre made by each end of the belt or rope in contact with the pulley rim. Alternative forms of this rim are shown in(b) to (d) Figure 11.1.
The socalled vee belt or rope (c) is now by far the most popular, having benefited from standardization and the resultant mass production by a number of reputable manufacturers. Circular crosssection ropes (d) are now rarely used for fan drives, but the flat belt (a) has shown some signs of a revival. Its reduced radial thickness compared with vee ropes means that centrifugal forces tending to make the belt(s) leave the pulley are minimized and high belt speeds (and therefore power transmitted) are possible. It should be noted that whilst the belt is flat, the rim of the pulleys used with it are in practice slightly “crowned”, since this has been found to help in maintaining the belt centrally on the pulley.
If the tension at one end of the belt is T2 and the tension T1 at the other end is increased gradually, then the belt will eventually start to slip bodily around the pulley rim. The value of T1 at which slip takes place will depend upon the values of T2, q and the coefficient of friction m between the belt and the rim.
Consider a short length mn of belt, which subtends and angle dq at the pulley centre.
Let T be the tension on the end m and T+ dT must be due to the friction between the length mn of the belt and the pulley rim, and it will depend upon the normal reaction between mn and the rim and the side of the groove for the sections (c) and (d). Let R be the radial reaction between the pulley rim and the length mn of
<b> (c) <d) 
(e) 
Figure 11.1 Diagrammatic view of pulley and belts or ropes 178 FANS & VENTILATION
Equ 11.2 
Belt or rope and let Rn be the normal reaction between each side of the groove and the side of mn for the sections (c) and (d). Then for section (b): 5T =xR Equ 11.1 And for sections (c) and (d): ST = 2xRn But for these sections the radial reaction R is the resultant of the two normal reactions Rn, so that R = 2Rn sin a and, substituting for Rn in terms of R, 
8T = ^5 = h1R 
Sin a 
= 0.653 and 
= 6.56 
= e 
Hi = 
The maximum effective tangential pull exerted by the belt or rope on the pulley rim is, in each case, given by the difference between ^ and T2. It may be expressed in terms of the tension Ti of the tight side, the magnitude of which is, of course, determined by the crosssection of the belt or rope and the allowable stress in the material. For the flat belt under the above conditions the effective tension 
0.25 sin 22.5° 


= cos ec a 
Where:
Equ 11.3
Sin a
It follows, therefore, that the friction between mn and the grooved rim is the same as that between mn and a flat rim, if the actual coefficient of friction is replaced by the virtual value
Sin a
In the plane or rotation of the pulley the three forces which act on mn are the tensions T and T + 5T on the ends m and n and the radial reaction R. Since mn is in equilibrium under this system of forces the triangle of forces may be drawn as shown in (e) of Figure 11.1.
From this triangle, since 50 and 8T are small, R sT. 50, and substituting this value of R in equation 11.1:
5T s nT50or ^ s nS0
If both sides of this equation are integrated between corresponding limits, then :
• t, DT
••• ioge^ = ne for the vee belt or rope belt, T=0.878T1
And for the circular section rope, T=0.848T.
It is clear from these figures that the use of a grooved pulley rim with a suitable vee or circular rope section enables the material to be employed more efficiently than where a flat rim is used.
So far it has been assumed that the pulley is stationary. If the pulley is mounted on a shaft, which is supported in bearings, then the effective tangential force exerted by the belt or rope on the pulley may be used to transmit powerfrom the belt or rope to the pulley and thence to the shaft. The power transmitted may be determined when the effective tension and the speed of the belt or rope are known. But when the belt or rope is in motion, the stresses in the material are not simply those which arise form the power transmitted. There is in addition the centrifugal stress due to the inertia of the belt or rope as it passes round the pulley rim. The magnitude of this stress may be determined as shown in the following section.
Centrifugal stress in a belt or rope
Referring to Figure 11.2, let r be the radius of the pulley, v the speed of the belt or rope, a the crosssectional area and w the weight of the belt or rope per unit length.
The weight of the short length mn which subtends to angle 50 at the pulley centre, is wr50 and the centrifugal force on mn is given by:
•50 
G 
F~r— 
Wr50 v wv 


69 
Similarly, if a rope of circular section is used with a groove angle of 45°, then 
Equ 11.4
As it stands this equation applied to the flat rim (b), but if m is substituted for n, it will apply equally well to the grooved rims (c) and (d).
It must be emphasized that equation 11.4 gives the limiting ratio of the tensions Tt and T2 when the belt or rope is just about to slip bodily round the pulley rim. The actual ratio of the tensions may have a lower value, but cannot have a higher value than this limiting ratio.
The limiting ratio is very much increased, for given values of n and 0, by using a grooved section. For instance if q is 165° and x is 0.25, the limiting ratio for the flat rim is given by:
T 0.25—
A=e 12 =2.054
T2
If a vee rope or belt is used with a groove angle of 40°, then
0.25
T 0.731—
= 0.731 and —e 12 =8.21
Sin 20°
This force acts radially outwards and, if the pulley rim is flat, the only possible way in which it can be resisted is by applying two forces Tc to the ends of mn. The short length of belt is in equilib
Wv G 
Wv G 
■ T„ = 
Equ 11.6 
Rium under these three forces and the triangle of forces may be drawn. From the triangle of forces Tc may be expressed in terms of Fc. Since 60 is small, Fc « Tc50 and substituting for Fc from the above equation:
= 60 = T, • 80
Equ 11.5
The stress per unit area of the belt or rope material due to the inertia is given by:
F =Jc=w v_
С
A a g
It should be particularly noticed that the centrifugal stress is independent of the radius of curvature of the path. It has been assumed so far that the rim of the pulley is flat and that the centrifugal inertia force therefore gives rise to a stress in the belt or rope material which is additional to the stresses caused by the tensions T! and T2.
If, however, the pulley rim is grooved as at (c) and (d) in Figure 11.1, it would appear at first sight that the centrifugal force may be either wholly or partly balanced by the friction between the sides of the belt or rope and the sides of the groove, in which case Fc will be either zero or will have a value less than that given by equation 11.6. But there are two other factors which have to be taken into account in this connection.
First, if the powertransmitted by the belt or rope is such that limiting friction exists in the tangential director i. e. if the belt or rope is just on the point of slipping bodily round the rim, there can be no friction force opposed to the centrifugal force. Since this condition of limiting friction rarely, if ever, exists in practice, there can be no doubt that the centrifugal stress in that part of the belt or rope, which is in contact wit the rim, will be less than the stress calculated from equation 11.6.
Secondly, and more importantly in any actual drive, the part of the belt or rope between the pulleys is not straight but hangs in a curve. The free parts of the belt must therefore be subjected to the centrifugal stress given by equation 11.6. Hence, there is not justification for the assumption which is sometimes made that the centrifugal stress in a belt or rope running on a grooved pulley is less than that in the same belt or rope when running on a flat pulley.
Posted in Fans Ventilation A Practical Guide